\newtheorem{lem}{Lemma}
% define a few physics shortcuts
-\newcommand{\U}[1]{\text{ #1}} % units shortcut
+\newcommand{\U}[1]{\textnormal{\ \bareU{#1}}} % units shortcut
+\newcommand{\bareU}[1]{\textnormal{#1}} % units without preceding value
\newcommand{\E}[1]{\ensuremath{\cdot 10 ^{#1}}} % exponent shortcut
\newcommand{\dg}{\ensuremath{^{\circ}}} % degree symbol ^o
+\newcommand{\celsius}{\dg\text{C}} % degrees Celsius
+\newcommand{\farenheit}{\dg\text{F}} % degrees Farenheit
\newcommand{\vect}[1]{\ensuremath{\mathbf{#1}}} % make vectors bold
\newcommand{\Ohm}{\ensuremath{\Omega}} % units of resistance